Download formula sheet

Math 152 Calculus Formula
b
1.
Substitution Rule

g (b )
f ( g ( x))  g ( x)dx 
a
2.
 f (u)du
g (a)
 f ( x) g ( x)
Integration by Parts
f ( x)G( x)   f ( x)G( x)dx
=
d
b
3.
Volume of revolutions
 A( x)dx
V =
or
V =
a
c
d
b
2
V =    f ( x) dx
Volume of revolution (disk)
 A( y)dy
or
V =   f ( y ) dy

a
Volume of revolution (cylindrical shells)
b
4.
Arc Length
L =

a
 dy 
1  
 dx 
c
V =
b
d
a
c
 2xf ( x)dx or V =  2yf ( y)dy
2
d
dx
or
Surface of revolution S =

 dx 
1   
 dy 
d
2
 2f ( x) 1   f ' ( x) dx or S =
 2g ( y )
a
c
2
b
6.
L=
c
b
5.
Average value of a function
fave =
2
1
f ( x)dx
b  a a
dy
1  g ' ( y )  dy
2
Definite Integral
b

n
f ( x)dx  lim  f ( xi* )  x. , where xi* [ xi 1 , xi ] is a sample point and
n 
a
i 1
ba
x 
. x0  a, xi  a  i  x, xn  b.
n
Trapezoidal Rule
x
 f ( x0 )  2 f ( x1 )  2 f ( x2 )  ...  2 f ( xn1 )  f ( xn )
Tn 
Error Bound (Trapezoidal Rule)
If f ( x)  K x  [a, b],
2
ET 
K (b  a)3
K (b  a)3
and EM 
2
12n
24n 2
Simpson’s Rule
x
 f ( x0 )  4 f ( x1 )  2 f ( x2 )  4 f ( x3 )...  2 f ( xn2 )  4 f ( xn1 )  f ( xn )
Sn 
3
Error Bound (Simpson’s
Rule)
If f ( 4) ( x)  K x  [a, b],
ES 
K (b  a)5
180n 4
Table of Indefinite Integrals
 kdx kx  C
n
 x dx 
x n 1
 C, n  1
n 1
x
x
 e dx e  C
x
 a dx 
 sin x dx   cos x  C
 sec x dx  tan x  C
2
x
1
 x dx  ln x  C
1
dx  tan 1 x  C
1
2
ax
C
ln a
 cos x dx  sin x  C
 csc x dx   cot x  C
2

1
1 x
2
dx  sin 1 x  C
 tan xdx  ln sec x  C
 sec xdx  ln sec x  tan x  C
Trigonometric Identities.
cos 2 x  sin 2 x  1
1  cos 2 x
2
1  cos 2 x
2
cos x 
2
sin 2 x 
sec2x = 1 + tan2x
 
 cos
2
2
 
 
cos   cos   2 sin
 sin
2
2
 
 
sin   sin   2 sin
 cos
2
2
cos   cos   2 cos
 